Ms. Ebbeler spent nearly two years building an online version of “Introduction to Ancient Rome” with a team of designers at the University of Texas at Austin, where she is an associate professor of classics. Most of the heavy lifting came during the last academic year, when one of her colleagues taught the course to hundreds of undergraduates while she coordinated behind the scenes.

The process was challenging and occasionally chaotic, she says, but her team learned a lot about how to make the online course work. Which is why she was frustrated when her bosses declined her request to teach the course next year.

“I am puzzled that I was not given a course to teach that I have invested hundreds of hours developing,” she wrote in a memorandum to her colleagues in March, “but so it goes.”

Orchestrating a smooth handoff of an online course is a relatively new challenge for Austin’s classics department. Elsewhere it has become normal for professors to relinquish courses they helped create, especially if those courses were built in a format that places content, not the instructor, at the forefront of the student experience.

For many institutions, online education has been an opportunity not only to increase the number of enrolled students, but also to focus on designing courses that are compelling no matter who is leading them.

“You’re seeing more and more of instructors rotating in and out of courses once they’re developed, because obviously the time to develop a course is a lot,” says John Haubrick, manager of instructional design at Pennsylvania State University’s online arm.

After being told she would not be leading her Online Rome course next year, Ms. Ebbeler learned that the instructor whom she had recommended to lead it instead — Steve Lundy, an adjunct faculty member who had taught the first two semesters of the course — had also been passed over for the job.

Frustrated, the professor vented on her blog. “I was stunned to learn yesterday, indirectly and in passing, that the classics chair had opted not to implement the ‘succession plan’ that I had carefully and thoughtfully crafted,” Ms. Ebbeler wrote earlier this month.

The instructors who were chosen to inherit the course, she added, “are either unqualified or underqualified for the specific tasks that the successful instruction of Online Rome requires.”

Reached by phone, Lesley A. Dean-Jones, chair of the classics department at Austin, said she would not comment on “personnel matters.” Asked if she would be willing to talk about the challenges relating to “succession” in online courses generally, Ms. Dean-Jones said no and hung up.

On the surface, Ms. Ebbeler’s Rome course seems to push the instructor to the margins. Students work through a series of online modules — containing course readings, links, and quizzes — on their own time. Mr. Lundy, the instructor, held review sessions on the campus every week, but they were optional and archived online. Students are required to show up in person only for examinations.

But Ms. Ebbeler says the role of the instructor remains crucial. “They think it doesn’t matter who they put in charge because the course will teach itself,” she says. “And yet I’ve been clear all along that that’s not the case.”

Rolando Garza, an instructional designer at Texas A&M University at Kingsville, says managing a course handoff can be challenging, but it helps if somebody who was involved in creating of the course works closely with the new instructor.

Mr. Garza says he sometimes plays that role at Kingsville, helping instructors find their bearings in courses he helped the original authors build. (This seems to be happening with the Online Rome course; Mr. Lundy says he will be working directly with the instructors who were picked to lead the course this summer and fall. “I feel quite involved at this point,” he says.)

But that does not mean there is not sometimes conflict if a professor who built a course does not trust that it will be well cared-for.

“That’s your baby,” says Mr. Garza. “You built it.” And entrusting it to a stranger can be hard.